Glossary

An actuator is a machine component that
converts a Simulink® signal into Simscape™ Multibody™ force, torque,
or motion signals.

You can configure a bodyactuator to
apply forces/torques to a body either as an explicit function of time
or through feedback forces/torques.

You can configure a joint actuator to
apply forces/torques between the bodies connected on either side of
the joint.

You can configure a driver actuator to
apply relative motion between the bodies connected on either side
of the driver.

Two specialized Simscape Multibody actuators set joint
initial conditions and apply stiction to
a joint, respectively.

Actuator

What the Actuator Does

Body Actuator

Applies forces to a body

Driver Actuator

Applies motion to a time-dependent constraint

Joint Actuator

Applies forces or motions to a joint

Joint Initial Condition Actuator

Sets a joint's initial conditions

Joint Stiction Actuator

Applies static and kinetic friction to joint motion

A Simscape Multibody Actuator block has an open round Simscape Multibodyconnector port for connecting with a Body, Joint, or Driver
block and an angle bracket > Simulink inport for connecting
with normal Simulink blocks, such as Source blocks for generating
force or torque signals.

A machine is in its assembled configuration once
it passes from its home configuration through its initial configuration
and its disassembled joints are then assembled. The assembly of joints
can change body and joint configurations.

An assembled joint restricts the Body coordinate
systems (CSs) on the two bodies at either end of the joint.

For an assembled prismatic joint, the
two Body CS origins must lie along the prismatic axis. The two Bodies
translate relatively along the same axis.

For an assembled joint with multiple prismatic primitives, the
two Body CS origins must lie in the plane or space defined by the
directions of the prismatic axes.

For an assembled revolute joint, the
two Body CS origins must be collocated. The two Bodies rotate relatively
about the same axis.

For an assembled joint with multiple revolute primitives, the
two Body CS origins must be collocated.

For an assembled spherical joint, the
two Body CS origins must be collocated at the spherical primitive's
pivot point. The two Bodies pivot relatively about this common origin.

You specify an assembly tolerance for assembled joints, the
maximum dislocation distance allowed between all pairs of assembled
Body CS origins and the maximum angle of misalignment between all
pairs of assembled Body motion axes. If the distance dislocations
and/or axis misalignments in an assembled joint grow larger than the
assembly tolerance, the simulation stops with an error.

An assembly represents a mechanical system
in computer-aided design (CAD). An assembly includes parts (bodies
with full geometric, mass, and inertia tensor information), as well
as constraints (sometimes called mates)
restricting the degrees of freedom of the parts.

Every assembly has a fundamental root attached
to the assembly coordinate origin. Assemblies can also have one or
more subassemblies branching off the main assembly
at a single point.

Assembly specifications typically also include design tolerances
and how subassemblies move and how they are connected to the main
assembly.

The assembly tolerance is a range that
determines how closely an assembled joint must
be collocated and aligned. An assembled joint is connected
on either side to Body coordinate systems (CSs) on two Bodies and
restricts the relative configurations and motions of those Body CSs.

The assembly tolerances set the maximum dislocation of
Body CS origins and maximum misalignment of motion
axes allowed in assembled joints during the simulation.

For assembled prismatic primitives, each
pair of Body CS origins must lie in the subspace defined by the prismatic
axes. Each pair of Bodies translates along these common axes.

For assembled revolute primitives, each
pair of Body CS origins must be collocated and their respective rotational
axes aligned. Each pair of Bodies rotates about these common axes.

For an assembled spherical primitive, the
pair of Body CS origins must be collocated. The two Bodies pivot about
this common origin.

A Simscape Multibody simulation attempts to assemble all joints
in your machine at the start of simulation, including initially disassembled
joints. If it cannot, the simulation stops with an error.

If the two Body CSs separate or the joint axes misalign in a
way that makes their connecting assembled joint primitives no longer
respect the assembly tolerances, the simulation stops with an error.

A body is the basic element of a mechanical
system. It is characterized by its

Mass properties (mass and inertia tensor)

Position and orientation in space

Attached Body coordinate systems

Bodies are connected to one another by joints, constraints,
or drivers. Bodies carry no degrees of freedom.

You can attach to a Body block any number of Body coordinate
systems (CSs). All Simscape Multibody Bodies automatically maintain
a minimum of one Body CS at the body's center of gravity (CG). The
Body block has distinctive axis triad CS ports , instead of the open, round connector
ports ,
to indicate the attached Body CSs.

A Body CS is a local coordinate system
(CS) attached to a body, carried along with that body's motion. In
general, bodies accelerate as they move, and therefore Body CSs define
noninertial reference frames.

You can attach any number of Body CSs to a Body block, and you
can choose where to place the Body CS origins and how to orient the
Body CS axes. The Body block has distinctive axis triad Body CS ports instead of the open, round connector
ports, to give you access to these Body CSs for connecting Joint,
Sensor, and Actuator blocks.

Every Body block has an automatic, minimum Body CS at its center
of gravity (CG). By default, it also has two other Body CSs for connection
to adjacent Joints. Once you set the origin and axis orientation of
each Body CS during Body configuration, the Body CSs are interpreted
as fixed rigidly in that body during the simulation.

The center of gravity or center of mass
of an extended body is the point in space about which the entire body
balances in a uniform gravitational field. For translational dynamics,
the body's entire mass can be considered as if concentrated at this
point.

Every Body block has an automatic, minimum Body coordinate system
(CS) with its origin at the CG — the CG CS. This origin point
and the Body CS coordinate axes remain fixed rigidly in the body during
the simulation.

A machine diagram contains one or more closed loops if,
beginning at a starting point, you can trace a path through the machine
back to the starting point without jumping out of or cutting the diagram.
The number of closed loops is equal to the minimum number of cuttings
needed to convert the diagram into a tree or open machine.

A composite joint is a joint compounded
from more than one joint primitive and thus representing more than
one degree of freedom. The joint primitives constituting a composite
joint are the primitives of that joint.

A spherical primitive represents three rotational degrees of
freedom, but is treated as a primitive.

Computer-aided design systems or platforms provide an environment
to design machines, with full geometric information about parts (bodies)
and their spatial relationships, as well as the degrees
of freedom and mass properties of the parts.

You connect each Simscape Multibody block to another by using Simscape Multibody connection
lines. These lines function only with Simscape Multibody blocks.
They do not carry signals, unlike normal Simulink lines, and
cannot be branched. You cannot link connection lines directly to Simulink lines.

By default, connection lines appear red and dashed if they are
not anchored at both ends to a connector port . Once you anchor them, the lines become black
and solid.

However, if you have selected a nonnull choice in Sample
Time Display from your model's Format menu,
the connection line displays instead with whatever color chosen to
indicate the sample time.

A connector port is an anchor for a connection line. Each Simscape Multibody block
has one or more open round Simscape Multibody connector ports for connecting to other Simscape Multibody blocks.
You must connect these round ports only to other Simscape Multibody round
ports. When an open connector port is attached to a connection line, the Port changes
to solid .

A Connection Port block is provided in
the Simscape Multibody library to create a round Simscape Multibody connector
port for an entire subsystem on that subsystem's boundary.

A constraint is a restriction among degrees
of freedom imposed independently of any applied forces/torques. A constraint removes
one or more independent degrees of freedom, unless that constraint
is redundant and restricts degrees of freedom
that otherwise could not move anyway. Constraints can also create inconsistencies with
the applied forces/torques that lead to simulation errors.

Constraints are kinematic: they
must involve only coordinates and/or velocities. Higher derivatives
of coordinates (accelerations, etc.) are determined by the Newtonian
force and torque laws and cannot be independently constrained.

Constraints are holonomic (integrable
into a form involving only coordinates) or nonholonomic (not
integrable; that is, irreducibly involving velocities).

Constraints specify kinematic relationships that are
explicit functions of time (rheonomic) or not (scleronomic).

In Simscape Multibody models with closed loops,
one Joint, Constraint, or Driver per loop is internally cut and replaced
by an invisible scleronomic or rheonomic constraint.

Constraints are redundant if
they independently impose the same restrictions on a machine.

In computer-aided design (CAD) assemblies, a constraint restricts
one or more degrees of freedom of the assembly parts. (CAD constraints
are sometimes called mates.) When a CAD assembly
is converted to a Simscape Multibody model, such restricted degrees
of freedom are translated into specific joints. (Simscape Multibody bodies
have no degrees of freedom.)

The convex hull of a set of points in space
is the surface of minimum area with convex (outward) curvature that
passes through all the points in the set. In three dimensions, this
set must contain at least four distinct, non-coplanar points to make
a closed surface with nonzero enclosed volume.

The convex hull is an option for visualizing a Simscape Multibody body.
The set of points is all the Body coordinate system (CS) origins configured
in that Body block. The visualization of an entire machine is the
set of the convex hulls of all its bodies. A Simscape Multibody convex
hull excludes the body's center of gravity CS.

If a Body has fewer than four distinct, non-coplanar Body CSs,
its convex hull is a lower-dimensional figure:

A coordinate system is a geometric object
defined, in a particular reference frame, by a choice of origin and
orientation of coordinate axes, assumed orthogonal
and Cartesian (rectangular). An observer attached to that CS measures
distances from that origin and directions relative to those axes.

Simscape Multibody software supports two CS types:

World: global or absolute
inertial CS at rest

The assembly origin of a CAD assembly translated into a model
becomes the World CS origin.

Local:

Grounded CS

Body CS, including the center of gravity (CG) CS

Constraint points on CAD parts in a CAD assembly translated
into a model become Body CS origins on the bodies representing the
parts.

A degree of freedom is a single coordinate
of relative motion between two bodies. Such a coordinate is free only
if it can respond without constraint or imposed motion to externally
applied forces or torques. For translational motion, a DoF is a linear
coordinate along a single direction. For rotational motion, a DoF
is an angular coordinate about a single, fixed axis.

The directionality of a joint, constraint,
or driver is its direction of forward motion.

The joint directionality is set by the order of the joint's
connected bodies and the direction of the joint axis vector. One body
is the base body, the other the follower body.
The joint direction runs from base to follower, up to the sign of
the joint axis vector. Reversing the base-follower order or the joint
axis vector direction reverses the forward direction of the joint.

Joint directionality sets the direction and the positive sign
of all joint motion and force-torque data.

Directionality of constraints and drivers is similar, except
there is no joint axis, only the base-follower sequence.

A driver is a constraint that restricts
degrees of freedom as an explicit function of time (a rheonomic constraint)
and independently of any applied forces/torques.
A driver removes one or more independent degrees of freedom, unless
that driver is inconsistent with the applied
forces/torques and forces a simulation error.

You specify the driver function of time in a dialog box in terms
of an input Simulink signal from a Driver Actuator.

The equivalent ellipsoid of a body is the
homogeneous solid ellipsoid, centered at the body's center of gravity,
with the same principal moments of inertia and principal axes as the
body. A homogeneous solid ellipsoid is the simplest body with three
distinct principal moments.

Every body has a unique equivalent ellipsoid, but a given homogeneous
ellipsoid corresponds to an infinite number of other, more complicated,
bodies. The rotational dynamics of a body depend only on its equivalent
ellipsoid (which determines its principal moments and principal axes),
not on its detailed shape.

The equivalent ellipsoid is an option for visualizing a Simscape Multibody body.

Euler angles mathematically represents
a three-dimensional spherical rotation as a product of three successive
independent rotations about three independent axes by three independent
(Euler) angles. Follow the Euler angle convention by

Rotating about one axis (which rotates
the other two).

Then rotating about a second axis (rotated
from its original direction) not identical to the first.

The fundamental root is a point in a computer-aided
assembly that does not move, usually coincident with the assembly
origin. All translational and rotational motion of parts in the assembly
reference this unmoving point.

The computer-aided assembly origin is recreated as the World
coordinate system origin in a CAD-generated model.

A grounded CS is a local CS attached to
a ground point. It is at rest in World, but its origin is wherever
the ground point is and in general shifted with respect to the World
CS origin. The coordinate axes of a grounded CS are always parallel
to the World CS axes.

The World coordinate axes are defined so that:

+x points right

+y points up (gravity in -y direction)

+z points out of the screen, in three dimensions

You automatically create a Grounded CS whenever you set up a
Ground block.

The bodies of a machine are in their home configuration when
they are positioned and oriented purely according to the positions
and orientations entered into the Body dialogs. This configuration
assumes zero body velocities.

The inertia tensor of a Simscape Multibody body is always evaluated
in that body's center of gravity coordinate system (CG CS). That is,
the origin is set to the body's CG and the coordinate axes are the
CG CS axes.

Because the CG CS of a Body block is fixed rigidly in the body
during simulation, the values of the inertia tensor components do
not change as the body rotates.

A machine is in its initial configuration once
all initial condition actuators have been applied to its joints. This
step can change the positions and orientations of the machine's bodies,
as well as apply nonzero initial velocities.

A joint is a machine component that represents
one or more mechanical degrees of freedom between two bodies. Joint
blocks connect two Body blocks in a Simscape Multibody schematic. A
Joint has no mass properties such as a mass or an inertia tensor,
and the reaction force and torque are equal and opposite on its two
connected Bodies.

A joint primitive represents one translational
or rotational degree of freedom or one spherical (three rotational
degrees of freedom in angle-axis form). Prismatic and revolute primitives
have motion axis vectors. A weld primitive has no degrees of freedom.

A primitive joint contains one joint primitive.
A composite joint contains more than one joint
primitive.

Joints have a directionality set by their base-to-follower Body
order and the direction of the joint primitive axis. The sign of all
motion and force-torque data is determined by this directionality.

A kinematic analysis of a mechanical system
specifies topology, degrees of freedom (DoFs), motions, and constraints,
without specification of applied forces/torques or the mass properties
of the bodies.

The machine state at some time is the set
of all

Instantaneous positions and orientations

Instantaneous velocities

of all bodies in the system, for both linear (translational)
and angular (rotational) DoFs of the bodies.

Specification of applied forces/torques and solution of the
system's motion as a function of time are given by the system's dynamics.

A local coordinate system (CS) is attached
to either a Ground or a Body:

A Grounded CS is automatically
defined when you represent a ground point by a Ground block and is
always at rest in the World reference frame. The origin of this Grounded
CS is the same point as the ground point and not in general the same
as the World CS origin.

You define one or more Body CSs when
you configure the properties of a Body. A Body CS is fixed rigidly
in the body and carried along with that body's motion.

To indicate an attached coordinate system, a Body block has
an axis triad CS port in place of the open, round connector port .

In a Simscape Multibody model, a machine is
a complete, connected block diagram representing one mechanical system.
It is topologically isolated from any other machine in your model
and has at least one ground and exactly one Machine Environment block.

A machine precision constraint implements
motion restrictions on constrained degrees of freedom to the precision
of your computer processor's arithmetic. It is the most robust, computationally
intensive, and slowest-simulating constraint.

The precision to which the constraint is maintained depends
on scale or the physical system of units.

A massless connector is a machine component
equivalent to two joints whose respective primitive axes are spatially
separated by a fixed distance. You can specify the gap distance and
the axis of separation. The space between the degrees of freedom is
filled by a rigid connector of zero mass.

You can disconnect an open machine diagram
into two separate diagrams by cutting no more than one joint.

Such machines can be divided into two types:

An open chain is a series of
bodies connected by joints and topologically equivalent to a line.

An open tree is a series of bodies
connected by joints in which at least one body has more than two joints
connected to it. Bodies with more than two connected joints define branch
points in the tree. A tree can be disconnected into multiple
chains by cutting the branch points.

You obtain the physical tree representation
from a full machine diagram by removing actuators and sensors and
cutting each closed loop once. The physical tree retains bodies, joints,
constraints, and drivers.

The inertia tensor of a body is real, symmetric, and diagonalizable,
with three real eigenvalues and three orthogonal eigenvectors. The principal
inertial moments or principal moments of inertia of
a body are these eigenvalues, the diagonal values when the tensor
is diagonalized.

The principal moments of a real body satisfy the triangle
inequalities: the sum of any two moments is greater than
or equal to the third moment.

If two of the three principal moments are equal, the body has
some symmetry and is dynamically equivalent to a symmetric
top. If all three principal moments are equal, the body
is dynamically equivalent to a sphere.

A quaternion mathematically represents
a three-dimensional spherical rotation as a four-component row vector
of unit length:

q=[nxsin(θ/2),nysin(θ/2),nzsin(θ/2),cos(θ/2)]=[qv,qs]

with q*q = 1. The vector n = (nx,ny,nz)
is a three-component vector of unit length: n·n =
1. The unit vector n specifies the
axis of rotation. The rotation angle about that axis is θ and follows the
right-hand rule.

The right-hand rule is the standard convention
for determining the sign of a rotation: point your right thumb into
the positive rotation axis and curl your fingers into the forward
rotational direction.

In CAD assemblies, the fundamental or subassembly root represents
a fixed point relative to which all part motion or subassembly part
motion is measured. The fundamental root is usually the same as the
assembly origin.

A rotation matrix mathematically represents
a three-dimensional spherical rotation as a 3-by-3 real, orthogonal
matrix R: RTR = RRT = I,
where I is the 3-by-3 identity and RT is
the transpose of R.

R=(R11R12R13R21R22R23R31R32R33)=(RxxRxyRxzRyxRyyRyzRzxRzyRzz)

In general, R requires three independent
angles to specify the rotation fully. There are many ways to represent
the three independent angles. Here are two:

You can form three independent rotation matrices R1, R2, R3,
each representing a single independent rotation. Then compose the
full rotation matrix R with respect to fixed coordinate
axes (like World) as a product of these three: R = R3*R2*R1.
The three angles are Euler angles.

You can represent R in terms of
an axis-angle rotationn =
(nx,ny,nz)
and θ with n·n = 1. The
three independent angles are θ and the two needed to orient n. Form the antisymmetric matrix:

A sensor is a machine component that measures
the motion of, or forces/torques acting on, a body or joint. A sensor
can also measure the reaction forces in a constraint or driver constraining
a pair of bodies.

A Simscape Multibody Sensor block has an open round Simscape Multibody connector
port for
connecting with a Body or Joint block and an angle bracket > Simulink outport
for connecting with normal Simulink blocks, such as a Sinks block
like Scope.

A stabilizing constraint implements motion
restrictions on constrained degrees of freedom by modifying the dynamics
of a system so that the constraint manifold is attractive, without
changing the constrained solution. This constraint solver type is
computationally the least intensive, the least robust, and the fastest-simulating.

The precision to which the constraint is maintained depends
on scale or the physical system of units.

Stereolithographic (STL) format is an open
file format for specifying the three-dimensional surface geometry
or shape of a body. STL format specifies surface geometry by linked
triangles whose edges and vertices are oriented by the right-hand
rule.

A subassembly root is a point in a computer-aided
design (CAD) subassembly that does not move relative to the assembly
point off of which it branches. All translational and rotational motion
of parts in the subassembly reference this unmoving point.

A tolerancing constraint implements motion
restrictions on constrained degrees of freedom only up to a specified
accuracy and/or precision.

This accuracy/precision is independent of any accuracy/precision
limits on the solver used to integrate the system's motion, although
constraints cannot be maintained to greater accuracy than the accuracy
of the solver.

The precision to which the constraint is maintained depends
on scale or the physical system of units.

Tolerancing constraints are moderately robust and moderately
intensive and execute at moderate speed. They are less intensive than
machine precision constraints, but computationally more intensive
than stabilizing constraints.

Tolerancing constraints are most useful in realistic simulation
of constraint slippage ("slop" or "play").

The topology of a machine diagram is the
global connectivity of all its elements. A diagram's elements are
its bodies, and its connections are its joints,
constraints, and drivers. Two topologies are equivalent if
you can transform one diagram into another by continuous deformations
and without cutting connections or joining elements.

An open machine diagram has no closed loops.

An open chain is topologically
equivalent to a line; and each body is connected to only two other
bodies, if the body is internal, or one other body if it is at an
end.

An open tree has one or more branch
points. A branch point is where an internal body is connected
to more than two joints. A tree can be disconnected into multiple
chains by cutting at the branch points.

A closed loop machine diagram has one or
more closed loops. The number of closed loops is equal to the minimum
number of joints, minus one, that must be cut to dissociate a diagram
into two disconnected diagrams.

An actual diagram can have one of these primitive topologies
or can be built from multiple primitive topologies.

In the Simscape Multibody environment, World is
a kinematic and geometric construct defining both the absolute inertial reference frame (RF) and
absolute coordinate
system (CS) in that RF. World has a fixed origin
and fixed coordinate axes that cannot be changed.

The World coordinate axes are defined so
that:

+x points right

+y points
up (gravity in -y direction)

+z points
out of the screen, in three dimensions

The assembly origin of a CAD assembly translated into a model
becomes the World CS origin.